Randomized Local Model Order Reduction

TL;DR

This paper introduces a randomized approach to construct local approximation spaces for model order reduction, achieving near-optimal convergence efficiently by leveraging randomized linear algebra and probabilistic error estimation.

Contribution

It proposes an adaptive randomized algorithm to efficiently approximate optimal local approximation spaces in model order reduction, reducing computational costs.

Findings

The randomized algorithm approximates transfer operator ranges with high accuracy.

The probabilistic error estimator is proven to be both efficient and reliable.

Numerical experiments validate the theoretical convergence and efficiency.

Abstract

In this paper we propose local approximation spaces for localized model order reduction procedures such as domain decomposition and multiscale methods. Those spaces are constructed from local solutions of the partial differential equation (PDE) with random boundary conditions, yield an approximation that converges provably at a nearly optimal rate, and can be generated at close to optimal computational complexity. In many localized model order reduction approaches like the generalized finite element method, static condensation procedures, and the multiscale finite element method local approximation spaces can be constructed by approximating the range of a suitably defined transfer operator that acts on the space of local solutions of the PDE. Optimal local approximation spaces that yield in general an exponentially convergent approximation are given by the left singular vectors of this transfer operator [I. Babu\v{s}ka and R. Lipton 2011, K. Smetana and A. T. Patera 2016]. However, the direct calculation of these singular vectors is computationally very expensive. In this paper, we propose an adaptive randomized algorithm based on methods from randomized linear algebra [N. Halko et al. 2011], which constructs a local reduced space approximating the range of the transfer operator and thus the optimal local approximation spaces. The adaptive algorithm relies on a probabilistic a posteriori error estimator for which we prove that it is both efficient and reliable with high probability. Several numerical experiments confirm the theoretical findings.

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